On Splitting Rings for Azumaya Skew Polynomial Rings

• 来源期刊: 代数集刊（英文版）
• 学科: 数学
• 关键词: skew polynomial rings Azumaya algebras Galois extensions splitting rings
• 年，卷（期）: 2001,（1）
• 页码范围: 25-32
• 页数: 8页
• 分类号: O1
• 语种: 英文

摘要

Let B be a ring with 1, p an automorphism of B of order n for some integer n, and B[x; p] the skew polynomial ring in x over B such that 1, x,x2,... ,xn-1 are independent over B and xn ∈ U(Bp), where Bp is the set of elements in B fixed under p and U(Bρ) is the set of all units in Bρ. Let -ρ be the inner automorphism of B[x; p] induced by x. Assume n is a unit in B. It is shown that, for a -ρ-Galois extension B[x; p] over (B[x; ρ])-ρ (resp., a DeMeyer-Kanzaki Galois extention B over Bp), B[x; ρ] is Azumaya if and only if (B[x; ρ])-ρ (resp.,B) is Azumaya, and some splitting rings of B[x; ρ], (B[x; ρ])-ρ and B coincide.